In 1962, John Selfridge discovered the Sierpinski number k = 78557, which is believed to be the smallest such number. The Sierpinski problem attempts to prove that it is, in fact, the smallest Sierpinski number. In 1976, Nathan Mendelsohn determined that the second provable Sierpinski number is the prime k = 271129. The prime Sierpinski problem attempts to prove that this is the smallest prime Sierpinski number.

Should both of these problems be solved, k = 78557 will be established as the smallest Sierpinski number, and k = 271129 will be established as the smallest prime Sierpinski number. However, this would not prove that k = 271129 is the second provable Sierpinski number. Since the prime Sierpinski problem is testing all prime k's for 78557 < k < 271129, all that's needed is to test the composite k's for 78557 < k < 271129. Thus, theextended Sierpinski problem is established.

As of April 3rd, 2018, there remain 10 composite k's for which no primes have been found. They are as follows: